WASHINGTON — The U.S. Senate Foreign Relations Committee could finalize and take up a resolution this week that would authorize military strikes in Syria.

Sen. Bob Corker, R-Tenn., told reporters Tuesday on Capitol Hill that he and Chairman Robert Menendez, D-N.J., have a general “understanding” about what must be included in a use-of-force measure.

The committee hopes to complete a draft resolution “in the next few days,” Corker said.

He would not close the door on the possibility that the full panel could mark up the resolution later this week.

Corker was among congressional and committee leaders who met this morning with President Obama at the White House.

Corker said whatever a final resolution looks like it must allow U.S. officials to deal with any “retaliatory” measures Syrian President Bashar al-Assad might take.

While Corker said he sees “no holes” in Obama’s rationale for launching cruise missiles to punish Assad for allegedly killing more than 1,000 people with chemical weapons.

But several Democratic lawmakers told reporters Tuesday they and their colleagues still have reservations.

The same is true of House Republicans. Senate Minority Whip John Cornyn, R-Texas, told reporters Tuesday the president “has not made a case.” He also said he and other Republicans are worried the White House’s plan of limited strikes “could be too limited,” and could end up helping Assad’s forces.

Sen. Jeff Flake, R-Ariz., said he finds Obama’s contention that the target list would be mostly the same in a few weeks “strange.” Flake would not say how he will vote, saying only: “I’m still listening.”

Still, the White House picked up some critical Republican allies on Tuesday, gaining the endorsement of two House Republicans as Speaker John Boehner, R-Ohio, and Majority Leader Eric Cantor, R-Va., added voices of support.

“This is something that the United States as a country needs to do,” Boehner said after meeting with the president and Congressional leaders at the White House.

After the meeting, Cantor released a statement confirming his support of the president’s decision to strike. He also expressed a desire to rework elements of the Authorization for the Use of Military Force document that the White House sent up the Hill over the weekend, saying that “while the authorizing language will likely change, the underlying reality will not.”

Cantor also invoked Iran and Hezbollah as critical factors in making his decision — much like the president and Secretary of State John Kerry did last week — saying that the Syrian civil war is also “a sectarian proxy war that is exacerbating tensions throughout the Muslim world. It is clear Iran is a principal combatant in this conflict, and its direct involvement is an integral part of Iran’s bid to establish regional hegemony. Were Assad and his Iranian patrons to come out on top it would be a strategic victory for Iran, embolden Hezbollah, and convince our allies that we cannot be trusted.”

Also stressing the need for action in order to deter other foes, Boehner added that other countries with nuclear and chemical weapons need to “understand we’re not going to tolerate this type of behavior.”

The meeting at the White House comes a full week before Congress is due back in town for the official start of its next session. Seated between Boehner and Minority Leader Rep. Nancy Pelosi, D-Calif., the president repeated the administration’s rationale for hitting Syria, reiterating that Syria’s willingness to use chemical weapons “poses a serious national security threat to the United States and to the region, and as a consequence Assad and Syria needs to be held accountable.”

He also broadened the scope of some of the administration’s original comments on the Aug. 21 chemical attack that killed over 1,000 civilians, moving past an original focus on moral revulsion, and stressing international norms.

“This norm against using chemical weapons ... is there for a reason” he said, not only to protect civilians but also because weapons components “can be transported to non-state actors” and be used against American allies. Failing to punish Syria “also sends a message that international norms around issues like nuclear proliferation don’t mean much.”

Pelosi also backed military strikes, calling the chemical attacks “behavior outside the circle of civilized human behavior and we must respond.” She did, however, stress that the American public needs to hear more of the intelligence that the government has gathered about the attacks.

Menendez, whom the White House will work closely with in crafting the Senate’s version of the resolution, told reporters “there is without question evidence that the Assad regime committed this act.

“I think that the outline of what the military action that would be contemplated is it’s appropriate to respond to the actions that Assad took,” Menendez said.

“It is significant at the end of the day to deter” future Assad regime uses of things like sarin gas, he added.

Senate Armed Services Committee Chairman Carl Levin, D-Mich., who also attended the White House meeting, reiterated his desire for Washington to do more to aid Syria’s rebels.

“I urged the president this morning that it is vitally important to facilitate the ability of the vetted opposition to go after the weapons systems the Assad regime uses to launch chemicals as well as the tanks and artillery that defend those weapons systems,” Levin said in a statement. “I am hopefully that we will do so.”